Let us define the linear operator \(\gamma \)  by letting
\( \gamma _{ij}= \gamma ^{\prime }_{ij} - \overline{\gamma ^{\prime }_{ji} }, \ \ \ \ \ \gamma ^{\prime }_{ij} = \sum _k T^j_{ik} \varphi _k \) 
where \(e\)  is any local unitary frame with \(\varphi \)  its dual coframe. Note that this \(\gamma \)  is twice of the \(\gamma \)  in [1]}. Denote by \(\theta ^s\) , \(\Theta ^s\)  the matrix of connection and curvature of \(\nabla ^s\)  under the frame \(e\) . Then we always have \(\theta ^s=\theta + \gamma \) . For a fixed point \(p\in M\) , we may choose \(e\)  so that \(\theta |_p=0\) . This leads to
\( \Theta ^s = d\theta ^s - \theta ^s \wedge \theta ^s = \Theta + d\gamma - \gamma \wedge \gamma .\) 
Taking the \((1,1)\) -part of the entry forms, we get
\( (\Theta ^s)^{(1,1)}_{k\ell }-\Theta _{k\ell } \ = \ \overline{\partial }\gamma ^{\prime }_{k\ell } - \partial \overline{\gamma ^{\prime }_{\ell k}} + \sum _r \big ( \gamma ^{\prime }_{kr} \overline{\gamma ^{\prime }_{\ell r} } + \overline{\gamma ^{\prime }_{rk } } \gamma ^{\prime }_{r \ell } \big ) \) 
at the point \(p\) . Note that at the point \(p\) , since \(\theta |_p=0\) , the structure equation gives \(\overline{\partial }\varphi =0\)  at \(p\) , while \(\partial \varphi _i = \frac{1}{2}\sum _{j,k=1}^n T^i_{jk} \varphi _j\wedge \varphi _k\)  at \(p\) . So by looking at the coefficients in front of the \(\varphi _i \wedge \overline{\varphi }_j\)  term in the above identity, we get
\( R^s_{i\overline{j}k\overline{\ell }} - R_{i\overline{j}k\overline{\ell }} \ = \ T^{\ell }_{ik, \,\overline{j}} + \overline{T^{k}_{j\ell , \,\overline{i}}} + \sum _r \big ( T^{r}_{ik} \overline{T^{r}_{j\ell }} - T^{\ell }_{ir} \overline{T^{k}_{jr}} \big ) .\) 
This completes the proof of the lemma.
